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EN
The classical Cayley-Hamilton theorem is extended to Drazin inverse matrices and to standard inverse matrices. It is shown that knowing the characteristic polynomial of the singular matrix or nonsingular matrix, it is possible to write the analog Cayley-Hamilton equations for Drazin inverse matrix and for standard inverse matrices.
EN
The notion of positive continuous-time time-varying Lyapunov system is introduced. Necessary and sufficient conditions for the positivity and the asymptotic stability of the systems are established. Sufficient conditions for reachability are given and the Cayley-Hamilton theorem is extended for time-varying Lyapunov systems. The considerations are illustrated by numerical examples.
3
Positive 2D hybrid linear systems
EN
A new class of positive hybrid linear systems is introduced. The solution of the hybrid system is derived and necessary and sufficient condition for the positivity of the class of hybrid systems are established. The classical Cayley-Hamilton theorem is extended for the hybrid systems. The reachability of the hybrid system is considered and sufficient conditions for the reachability are established. The considerations are illustrated by a numerical example.
4
Generalizations of the Cayley-Hamilton theorem with applications
EN
New generalizations of the classical Cayley-Hamilton theorem for rectangular matrices, block matrices, matrices depending on parameters, discrete-time and continuous-time systems with delays, polynomial matrices with commuting matrices, n-D polynomial matrices, singular systems, right and left inverse of polynomial matrices, rational matrices, impulse response matrices and nonlinear time-varying systems are presented. Some applications of the generalizations and illustrating examples are also given.
PL
W pracy podano nowe uogólnienia klasycznego twierdzenia Cayley-Hamiltona na: macierze prostokątne, macierze blokowe, macierze zależne od parametrów, macierze układów ciągłych i dyskretnych z opóźnieniami, macierze wielomianowe z macierzami przemiennymi, macierze wielomianowe o elementach będącymi funkcjami n zmiennych, macierze układów singularnych, prawe i lewe odwrotności macierzy wielomianowych, macierze wymierne, macierze odpowiedzi impulsowych oraz na macierze układów nieliniowych o parametrach zmiennych w czasie. Podano również pewne zastosowania tych uogólnień twierdzenia Cayley-Hamiltona. Rozważania zostały zilustrowane przykładami.
5
An extension of the Cayley-Hamilton theorem for nonlinear time-varying systems
EN
The classical Cayley-Hamilton theorem is extended to nonlinear time-varying systems with square and rectangular system matrices. It is shown that in both cases system matrices satisfy many equations with coefficients being the coefficients of characteristic polynomials of suitable square matrices. The proposed theorems are illustrated with numerical examples.
PL
Klasyczne twierdzenie Cayleya-Hamiltona zostało uogólnione na prawe i lewe odwrotności macierzy wielomianowych oraz na macierze wymierne. Wykazano, że uogólnione twierdzenie Cayleya-Hamiltona jest prawdziwe również dla odwrotnej macierzy wymiernej. Wyznaczono zależności wiążące macierze spełniające uogólnione twierdzenie Cayleya-Hamiltona macierzy wymiernej i jej odwrotności.
EN
The classical Cayley-Hamilton theorem is extended forrightand left inverses of polynomial matrices and for rational matrices. It is shown that the extended Caley-Hamilton theorem is also valid for inverses of rational matrices. Relationships between matrices satisfying the extended Cayley-Hamilton theorem for rational matrices and their inverses are derived.
EN
The classical Cayley-Hamilton theorem is extended to continuous-time linear systems with delays. The matrices $A_0, A_1, dots, A_h in R^{n times n}$ of the system with $h$ delays $dot xleft(t right) = A_0 xleft(t right) + sum_{i = 1}^h {A_i xleft( {t - hi} right) + Buleft( t right)}$ satisfy $nh + 1$ algebraic matrix equations with coefficients of the characteristic polynomial $pleft( {s,w}right) = det left[ {I_n s - A_0 - A_1 w - cdots - A_h w^h }right]$, $w = e^{- hs}$.
8
An extension of the Cayley-Hamilton theorem for a standard pair of block matrices
EN
The Cayley-Hamilton theorem is extended for a standard pair of matrices partitioned into blocks that commute in pairs. The Victoria theorem (Victoria, 1982) is a particular case for E = I of the extended Cayley-Hamilton theorem. The new theorem is illustrated by an example. Some remarks on extension of the theorem for non-square block matrices are also given.
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